Discussion Paper
Deutsche Bundesbank
No 43/2013
Disentangling economic recessions
and depressions
Bertrand Candelon
(Maastricht University)
Norbert Metiu
(Deutsche Bundesbank)
Stefan Straetmans
(Maastricht University)
Discussion Papers represent the authors‘ personal opinions and do not
necessarily reflect the views of the Deutsche Bundesbank or its staff.
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Non-technical summary
The business cycle is traditionally described as a sequence of recessions and expansions
in aggregate economic activity. However, from a historical perspective it is well-known
that some recessions are much longer, deeper or more abrupt than others. Thus, the
question arises whether the binary characterization of the cyclical economic development
is not overly simplistic. For example, a semantic distinction between economic recessions
and “depressions” is commonly made in the economic policy discourse. Indeed, vast
contractions of real output have stimulated interest in rare “economic disasters” among
academic economists. The binary framework does not allow for considering such rare and
severe events by construction.
Questioning the validity of the binary approach constitutes the starting point of this
paper. We distinguish economic depressions and booms from ordinary recessions and
expansions using a novel nonparametric test. Given the peak and trough dates of the
U.S. business cycle between 1919 and 2009 provided by the National Bureau of Economic
Research (NBER), we compute characteristics which capture the length, depth and shape
of each recession and expansion phase. Business cycle phases are then classified into
four regimes according to the magnitude of these characteristics, such that we can also
distinguish between recessions vs. depressions and expansions vs. booms. Our method
identifies three economic depressions and one economic boom in the U.S. business cycle,
which coincide with important historical episodes, including the Great Depression and
the industrial production boom during the Second World War. Interestingly, despite
all comparisons between the Great Depression and the recent financial and economic
crisis, the so-called “Great Recession”, dated between December 2007 and June 2009
by the NBER, does not qualify as extraordinary in comparison with previous economic
downturns. Even though it is the most severe postwar recession, it is dwarfed along every
dimension by the prewar depressions.
In the second part of the paper, we analyze by means of logistic regressions whether the
four business cycle regimes can be predicted using macroeconomic and financial variables.
Numerous studies have shown that the slope of the yield curve exhibits good predictive
power for future recessions. The related literature also attributes some predictive power
to stock market returns, the growth rate of real output and the inflation rate. Is it
possible to tell whether the economy is heading toward a recession or a depression using
these leading indicators? Our results show that although the slope of the yield curve
outperforms other leading indicators in predicting recessions, it does not have anything
to say about future economic depressions. In contrast, stock market returns, real output
growth, and the inflation rate convey statistically relevant information for predicting
economic depressions: a drop in these variables signals a higher likelihood of a future
depression.
Nicht-technische Zusammenfassung
Der Konjunkturzyklus wird traditionell als Abfolge von Rezessionen und Expansionen der
gesamtwirtschaftlichen Aktivität beschrieben. Aus der historischen Perspektive gemeinhin bekannt ist jedoch die Tatsache, dass einige Rezessionen viel länger, tiefgreifender und
abrupter sind als andere. Daraus erwächst die Frage, ob die binäre Charakterisierung der
Schwankungen in der Wirtschaftsentwicklung nicht allzu simplistisch ist. Semantisch wird
beispielsweise im wirtschaftspolitischen Diskurs gewöhnlich zwischen wirtschaftlicher Rezession und Depression“ unterschieden. In der Tat haben schwerwiegende Kontraktionen
”
der realen gesamtwirtschaftlichen Produktion das Interesse der Wirtschaftswissenschaftler an seltenen wirtschaftlichen Desastern“ erregt. Der binäre Rahmen lässt es aufgrund
”
seiner Konstruktionsweise nicht zu, solch seltene und einschneidende Ereignisse gesondert
unter die Lupe zu nehmen.
Die Hinterfragung der Validität des binären Ansatzes stellt den Ausgangspunkt dieses
Forschungspapiers dar. Wir unterscheiden große wirtschaftliche Depressionen und Boomphasen von gewöhnlichen Rezessionen und Aufschwüngen im Konjunkturzyklus mithilfe
eines neuen nichtparametrischen Testverfahrens. Anhand von Daten des National Bureau
of Economic Research (NBER) zu den Höhe- und Tiefpunkten der Wirtschaftszyklen
in den Vereinigten Staaten zwischen 1919 und 2009 berechnen wir Kenngrößen, welche
Länge, Tiefe und Ausformung jeder Rezessions- und Aufschwungsphase erfassen. Die Phasen im Wirtschaftszyklus werden dann entsprechend der Dimension dieser Kenngrößen in
vier Ausprägungen unterteilt, um zwischen Rezession und Depression bzw. Aufschwung
und Boom zu differenzieren. Mit unserer Verfahrensweise identifizieren wir drei wirtschaftliche Depressionen und einen Boom im US-Konjunkturzyklus, die mit wichtigen historischen Zeitabschnitten, so der Großen Depression und dem Industrieboom während des
zweiten Weltkriegs, zusammenfallen. Interessant dabei ist die Beobachtung, dass sich ungeachtet aller Vergleiche der Großen Depression mit der letzten Finanz- und Wirtschaftskrise die sogenannte Große Rezession“, die das NBER auf die Zeitspanne von Dezember
”
2007 bis Juni 2009 datiert, in der Gegenüberstellung mit vorangegangen Konjunkturabschwüngen nicht als außergewöhnlich beweist. Obgleich es sich dabei um die schwerste
Rezession der Nachkriegszeit handelt, wird sie von den Depressionen der Vorkriegszeit in
jeder Dimension um Längen überragt.
Im zweiten Teil des Aufsatzes analysieren wir mittels logistischer Regressionen, ob die
bei einer solchen Charakterisierung der wirtschaftlichen Schwankungen identifizierten vier
Regime unter Zuhilfenahme von Makroökonomischen- und Finanzvariablen vorhersagbar
sind. Zahlreiche Studien haben gezeigt, dass das Gefälle der Zinsstrukturkurve gute Prognoseeigenschaften für kommende Rezessionen aufweist. Eine gewisse Prognosekraft misst
die einschlägige Fachliteratur auch den Aktienmarktrenditen, der Wachstumsrate der realen Wirtschaftsleistung und der Inflationsrate bei. Ist es mithilfe dieser Frühindikatoren
möglich zu erkennen, ob die Wirtschaft auf eine Rezession oder eine Depression zusteuert? Gemäß unseren Ergebnissen besitzt der Verlauf der Zinsstrukturkurve - trotz deren
Überlegenheit gegenüber anderen Frühindikatoren bei der Vorhersage von Rezessionen
- keinerlei Aussagekraft mit Blick auf heraufziehende Wirtschaftsdepressionen. Demgegenüber liefern Aktienmarktrenditen, das reale Wirtschaftswachstum und die Inflationsrate statistisch relevante Informationen für die Prognose wirtschaftlicher Depressionen:
ein Abfall dieser Variablen signalisiert eine höhere Wahrscheinlichkeit, dass in näherer
Zukunft eine schwerwiegende Depression im Anmarsch ist.
Bundesbank Discussion Paper No 43/2013
Disentangling Economic Recessions and Depressions∗
Bertrand Candelon
Maastricht University
Norbert Metiu
Deutsche Bundesbank
Stefan Straetmans
Maastricht University
Abstract
We propose a nonparametric test that distinguishes “depressions” and “booms”
from ordinary recessions and expansions. Depressions and booms are defined as
coming from another underlying process than recessions and expansions. We find
four depressions and booms in the NBER business cycle between 1919 and 2009,
including the Great Depression and the World War II boom. Our results suggest
that the recent Great Recession does not qualify as a depression. Multinomial
logistic regressions show that stock returns, output growth, and inflation exhibit
predictive power for depressions. Surprisingly, the term spread is not a leading
indicator of depressions, in contrast to recessions.
Keywords: Business cycles, Depression, Leading indicators, Multinomial logistic
regression, Nonparametric statistics, Outlier.
JEL classification: C14, C35, E32.
Contact addresses: Bertrand Candelon: Department of Economics, School of Business and
Economics, Maastricht University, PO Box 616, 6200 MD, Maastricht, The Netherlands. E-Mail:
[email protected]. Norbert Metiu: Corresponding author. Research Centre, Deutsche
Bundesbank, Wilhelm-Epstein-Straße 14, 60431 Frankfurt am Main, Germany. Tel.: +49 69 9566 8513.
E-Mail: [email protected]. Stefan Straetmans: Department of Finance, School of Business and Economics, Maastricht University. E-Mail: [email protected]. The authors
thank Ben Craig, Heinz Herrmann, Malte Knueppel, Christian Matthes, Guido Schultefrankenfeld, and
seminar participants at the Deutsche Bundesbank for helpful comments. Discussion Papers represent the
authors’ personal opinions and do not necessarily reflect the views of the Deutsche Bundesbank or its
staff.
∗
1
Introduction
The business cycle is traditionally modeled as a sequence of recessions and expansions
in aggregate economic activity. This binary approach characterizes the regime-switching
models of, inter alia, Hamilton (1989) and Potter (1995), as well as the turning-points
methodology of Bry and Boschan (1971), King and Plosser (1994) and Harding and Pagan
(2002). Moreover, the prediction of business cycles has typically also been approached via
binary models, see, e.g., Estrella and Mishkin (1998) and Kauppi and Saikkonen (2008).
However, some recessions and expansions have been much stronger than others – e.g., in
terms of duration or output growth – throughout the history of the U.S. business cycle.
This notion is also reflected by a semantic difference between recessions vs. “depressions”
and expansions vs. “booms” often made by policymakers and the popular press. One
possibility is that such extreme episodes are tail realizations from the same data generating
process (DGP) as ordinary recessions or expansions, which would imply that a two-regime
approach delivers correctly specified business cycle models. Alternatively, depressions and
booms may be considered as “outliers” arising from a different DGP than recessions and
expansions. Neglecting this multi-phase character of the business cycle may distort model
specifications of the true DGP.
The objective of this paper is therefore to investigate the possibility that the DGP of
the business cycle is characterized by more than two regimes. We propose a novel nonparametric outlier detection framework to distinguish economic depressions and booms
from ordinary recessions and expansions based on the occurrence of outliers in some popular business cycle characteristics.1 The considered characteristics include the duration,
amplitude, cumulated movements, and excess cumulated movements suggested by Harding and Pagan (2002). We classify a phase as depression or boom if at least one of its
characteristics exhibits an outlier for that particular time frame relative to other historical episodes. Our outlier detection algorithm results in a four-regime classification of the
U.S. business cycle. This enables us to refine some of the conclusions drawn from previous
recession prediction exercises conducted within the two-regime framework.
The amount of U.S. economic recessions and expansions over the past century is relatively small and the closed form of the statistical distribution of business cycle characteristics is unknown. We therefore propose a distribution-free outlier detection test
that combines two results from nonparametric statistics and that performs well in small
samples. Our method involves bootstrapping the empirical distribution of the business
cycle characteristics and computing the difference between the arithmetic mean and the
trimmed mean of each bootstrap sample. This yields a measure of central tendency termed
“mean-trimmed mean”. Singh and Xie (2003) show that the mean-trimmed mean displays
a multimodal histogram if the original sample contains one or more outliers. We employ
the Silverman (1981) test to assess the null hypothesis of the histogram’s unimodality.
If the histogram is multimodal, we sequentially remove the most extreme observations
from the sample until we end up with a unimodal histogram that is free of outliers. The
omitted observations correspond to the outlying business cycle characteristics. Applying
this statistical procedure, we identify four depressions and booms of the U.S. business
1
A multitude of outlier definitions co-exist. We adopt the definition that an outlier constitutes an
observation that does not arise from the same DGP as the majority of observations in the data set,
conform to Barnett and Lewis (1994).
1
cycle over the time horizon 1919 to 2009. These coincide with economically meaningful
episodes, like the Great Depression and the industrial production boom during the Second
World War. However, despite all comparisons between the Great Depression and the recent financial and economic crisis, our results suggest that the 2007-2009 Great Recession
does not qualify as a depression compared to previous historical episodes.
There is a growing interest in severe economic contractions in the macroeconomic
literature (see, e.g., Kehoe and Prescott, 2002). Large and infrequent economic slumps
may require a bolder set of policy interventions than ordinary recessions (see Eggertsson
and Krugman, 2012). At the same time, rare “economic disasters”, such as depressions
or wars, have been shown to play a role in determining asset risk premia, see, e.g., Barro
(2006, 2009), Gabaix (2012), and Wachter (2013). A handful of papers have tried to
incorporate such severe episodes into non-linear regime-switching time series models with
more than two possible business cycle states, including Tiao and Tsay (1994), Sichel
(1994), and Cakmakli, Paap, and van Dijk (2013). Meanwhile, substantial effort has been
invested in testing for the actual number of regimes within Markov-switching models,
see, e.g., Cho and White (2007), Carter and Steigerwald (2012), and Carrasco, Hu, and
Ploberger (2013). To the best of our knowledge, we are the first to identify rare and severe
recessions by means of nonparametric outlier detection techniques. Previous studies that
deal with outliers in macroeconomic time series include, e.g., Balke and Fomby (1994)
and Giordani, Kohn, and van Dijk (2007), but these are parametric in nature.
From the preceding literature it is well-known that financial and macroeconomic variables, such as the slope of the yield curve, stock market returns, or real output growth,
exhibit some predictive power for future recessions and expansions (see, e.g., Harvey,
1988; Estrella and Hardouvelis, 1991; Estrella and Mishkin, 1998; Birchenhall, Jessen,
Osborn, and Simpson, 1999; Hamilton and Kim, 2002; Kauppi and Saikkonen, 2008;
Rudebusch and Williams, 2009; Christiansen, 2013). The question arises to what extent
these leading indicator properties carry over to the four-regime business cycle classification
that emerges from applying our outlier detection test. The predictive ability of financial
and macroeconomic variables is of potential importance for policymakers who would like
to distinguish between an impending recession and a rare economic disaster. Thus, one
would like to determine whether traditional leading indicators of recessions and expansions exhibit a different information content for depressions and booms. In fact, upon
applying a multinomial logistic regression model to the four-regime business cycle, we
find that the slope of the yield curve preserves its predictive power towards recessions but
its leading indicator property vanishes for economic depressions. However, we are able to
show that variables like past real output growth, inflation, and stock market returns help
to predict extraordinary business cycle fluctuations.2
The remainder of the paper is organized as follows. Section 2 starts with a definition
of business cycles and their characteristics. Subsequently, we introduce the nonparametric
outlier test. Finally, we report estimated cycle characteristics across the NBER business
cycle between 1919 and 2009, and we present the results from the outlier detection procedure. Section 3 provides a short theoretical digression on multinomial logit regressions.
Next, the section compares empirical results of binomial with multinomial logit specifica2
Throughout the paper “prediction” refers to in-sample predictive ability, since out-of-sample forecasts
of depressions and booms would not make sense given the scant amount of genuine depression and boom
phases within the historical sample.
2
tions and univariate with multivariate logit specifications, respectively. Finally, Section 4
provides a summary and conclusions.
2
2.1
The four-regime business cycle
Characterizing business cycles
Classical studies define business cycle fluctuations by means of turning points (peaks and
troughs) in the level of real economic activity (see, e.g., Burns and Mitchell, 1946; Bry
and Boschan, 1971; King and Plosser, 1994; Harding and Pagan, 2002). A complete cycle
in logarithmic real output yt consists of a recession phase from a peak to the subsequent
trough and an expansion phase from a trough to the subsequent peak. We use the
National Bureau of Economic Research (NBER) peaks and troughs in economic activity
to construct a binary variable Stbin that either reflects a recession phase (Stbin = 1) or an
expansion phase (Stbin = 0). The turning point dates published by the NBER represent a
consensus chronology of the U.S. business cycle.
To assess the relative strength of recessions and expansions, we focus on the unconditional frequency distribution of four commonly used business cycle characteristics: the duration (D), amplitude (A), cumulative movements (C) and excess cumulative movements
(E) computed for each business cycle phase. Formal definitions of these characteristics
are presented in the appendix. Previous work that characterizes the cyclical behavior of
real economic activity by means of these business cycle characteristics include, e.g., Harding and Pagan (2002) and Camacho, Perez-Quiros, and Saiz (2008). Bordo and Haubrich
(2010) have employed these characteristics to analyze cycles in money, credit, and output,
while others have used the same measures to investigate financial cycles (see, e.g., Pagan
and Sossounov, 2003; Claessens, Kose, and Terrones, 2012).
2.2
Testing for depressions and booms
In order to distinguish depressions and booms from ordinary business cycle phases, we
apply an outlier testing procedure to the previously described business cycle characteristics. A given recession (expansion) episode is classified as an economic depression (boom)
if at least one of its four characteristics is an outlier for that period. More precisely,
a depression (boom) is identified as follows. Provided that n recessions (expansions)
are observed, we have a sample for each recession (expansion) characteristic denoted by
X1 , ..., Xn (Xi stands for either Di , Ai , Ci , or Ei ). The observation Xi that corresponds to
phase i = 1, ..., n can be seen as a random draw from an unknown cumulative distribution
function FX (.). We define phase i as a depression (boom) if Xi is an outlier with respect
to the distribution FX (.), i.e., Xi is not distributed according to FX (.).
We propose a distribution-free test for the null hypothesis that the sample X1 , ..., Xn is
free of outliers. The test combines two established results from nonparametric statistics.
First, Singh and Xie (2003) introduce a “Bootlier Plot” to graphically detect the presence
of outliers in a data set. The Bootlier Plot is the bootstrap density plot of the sample
mean-trimmed mean statistic and its multimodality reflects the presence of outliers in the
sample. Second, Silverman (1981) proposes a distribution-free test for the unimodality of
a probability density function. Hence, we apply Silverman’s test to Bootlier Plots of phase
3
characteristics in order to detect outliers in these characteristics. If the null hypothesis
of unimodality is rejected for the full sample, we proceed with ordering the observations
X1 , ..., Xn into ascending order, sequentially dropping observations from the tails of the
ordered sample, and repeating Silverman’s test on Bootlier Plots of these shrinking subsamples. We continue the iterative omission of the most extreme observations until the
(subsample) Bootlier Plot becomes unimodal. The deleted observations can be identified
as outliers.3
Let us now discuss this nonparametric outlier detection procedure in somewhat more
detail. The Bootlier Plot is obtained as follows. Let X1b , ..., Xnb (b = 1, 2, ..., B) denote
bootstrapped samples of size n based on the original sample X1 , ..., Xn of a particular
business cycle characteristic. The resampling is repeated B = 10, 000 times. The meantrimmed mean (MTM) statistic is defined as the difference between the arithmetic mean
and the k-trimmed mean of the bth bootstrap sample:
MT M b =
n
n−k
X
1X b
1
b
X(i)
,
Xi −
n i=1
n − 2k i=k+1
(1)
b
where X(i)
are the ascending order statistics and k is a trimming value. Upon assuming
that FX (.) exhibits finite first and second moments, the central limit theorem applies and
the pdf of the mean-trimmed mean fM T M (.) converges asymptotically to a standard normal distribution in the absence of outliers. Singh and Xie (2003) show that fM T M (.) can
be expressed as a multimodal mixture of normal densities if the original sample X1 , ..., Xn
contains at least one outlier. The separation between the normal mixing components
arises because only some of the bootstrap samples contain the outliers. As a result,
fM T M (.) exhibits one mode associated with the distribution FX (.) and at least another
mode corresponding to one or more outliers.4
In order to test the null hypothesis of unimodality of fM T M (.) (absence of outliers in
X1 , ..., Xn ) against the alternative hypothesis of multimodality (presence of one or more
outliers), we apply the test proposed by Silverman (1981). The test uses as an input the
kernel density estimate of the density function fM T M (.). The kernel density estimator of
the MTM statistic at any point x can be expressed as:
B
1 X
x − MT M b
f (x, h) =
K(
),
Bh b=1
h
(2)
where h is a bandwidth and K(.) is a kernel function chosen to be the standard normal
density function following Silverman (1981). For a large class of kernel functions including
the standard normal, the number of modes of f (x, h) decreases as the bandwidth h is
increased. Thus, a sufficiently large h exists, for which the kernel density f (x, h) has a
single mode in the interior of a given closed interval ℑ. The narrowest bandwidth for which
3
By definition, outliers must be located in the upper or lower tails of the ascending order statistics X(1) , X(2) , ...,X(n−1) , X(n) . We sequentially cancel the most extreme observations by considering
the subsamples: (X(1) , ..., X(n−1) ), (X(2) , ..., X(n) ), (X(1) , ..., X(n−2) ), (X(2) , ..., X(n−1) ), (X(3) , ..., X(n) ),
(X(1) , ..., X(n−3) ), etc. We stop this process once unimodality can no longer be rejected.
4
As recommended by Singh and Xie (2003), we compute the MTM statistic with a trimming value of
k = 2. Singh and Xie (2003) show that in the presence of an outlier the separation between the modes
of the bootstrap density fMT M (.) is approximately proportional to 1/k independent of the sample size.
4
1.4
3.5
1.2
3.0
kernel density
kernel density
Figure 1: Bootlier Plots of NBER Recession Durations
1.0
0.8
0.6
0.4
0.2
2.5
2.0
1.5
1.0
0.5
0.0
-0.8 -0.4
0.0
0.4
0.8
1.2
1.6
2.0
0.0
-2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 2.0
2.4
bootstrap MTM statistics
bootstrap MTM statistics
(a) Durations of all recessions
(b) Durations without Great Depression
Note: Bootlier Plots of durations for NBER business cycle recessions between March 1919 and June
2009 (full sample) and for the subsample that excludes the duration of the Great Depression (40
months).
the kernel density estimate is unimodal is called the “critical bandwidth”. Intuitively, the
critical bandwidth of a multimodal density should be larger than that of a unimodal
density because a larger bandwidth is required to smooth out the multiple modes. This
provides a rationale for using the critical bandwidth as test statistic.
To implement Silverman’s test, we start with estimating the density of the MTM
statistic in Equation (1) using the kernel density estimator in Equation (2). Next, we
estimate the critical bandwidth ĥcrit . Let us denote the kernel density estimated with this
bandwidth as fˆ(., ĥcrit ). In order to determine the small sample distribution of the critical
bandwidth ĥcrit , we draw 1,000 bootstrap samples from the kernel density fˆ(., ĥcrit ). For
each draw, the density of the bootstrapped MTM statistic is again estimated using the
kernel density estimator defined in Equation (2). Let ĥ∗crit denote the critical bandwidth
of the kernel density
obtainedfor one bootstrap draw. The null hypothesis of unimodality
is rejected if P r ĥ∗crit ≤ ĥcrit ≥ 1 − α, where α is the nominal size.
If unimodality of the full-sample Bootlier Plot is rejected, we calculate Bootlier Plots
and perform Silverman’s test on subsamples by eliminating the most extreme observations
from the tails of the full sample X1 , ..., Xn . We continue shrinking the sample until the
null of unimodality can no longer be rejected.
Further details on the nonparametric outlier detection procedure, including its finite
sample behavior, are reported in a companion paper (see Candelon and Metiu, 2013).
The latter paper, inter alia, shows that the size and power properties of the outlier test
are satisfactory in small samples like the ones encountered in this paper.
As a simple example of how our outlier testing procedure works, consider the durations of NBER business cycle recessions. Figure 1 (a) and Figure 1 (b) report the Bootlier
Plots for the full sample and the subsample excluding the duration of the Great Depression
phase, respectively. The multimodal plot implies that the Great Depression’s duration is
an outlier of the duration’s empirical distribution. This suggests that the 1929-33 down5
Figure 2: The U.S. Business Cycle
4.8
4.4
log real output
4.0
3.6
3.2
2.8
2.4
2.0
1.6
1.2
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010
Note: Log levels of U.S. real industrial production, March 1919 - December 2010. NBER recessions
are shaded in grey.
turn was a genuine depression episode. Upon removing this data point from the historical
duration sample, the subsample Bootlier Plot becomes unimodal and Silverman’s test
does no longer reject the null of unimodality (p-value of 0.22). This indicates that there
are no other outliers left in the empirical distribution of recession durations.
2.3
Depressions and booms in the U.S. business cycle
We compute the duration, amplitude, cumulated movements, and excess cumulated movements of the U.S. business cycle using the NBER turning points. Figure 2 plots the log
of monthly U.S. real industrial production output between March 1919 and December
2010 with NBER recessions shaded in grey. Table 1 reports values of the characteristics computed for each recession and expansion phase. While on average recessions last
about 10 months, the longest recession lasted 40 months from the peak in August 1929
till the trough in March 1933. The severity of recessions also varies significantly across
different episodes: the average peak-to-trough amplitude of a recession is -17.6%, while
the difference between the amplitude of the most severe (August 1929 - March 1933) and
the mildest (March 2001 - November 2001) recession is 68.6 percentage points. Similar
patterns emerge for expansions: the difference between the shortest and longest expansion
is roughly 9 years, while the trough-to-peak output gains vary between 5.9% and 99.5%
over the analyzed period. The varying strength of contractions and expansions suggests
that one may gain more insight into the nature of the business cycle by further refining
its traditional two-regime classification.5
Figure 3 shows the kernel density estimates of the Bootlier Plots corresponding to each
5
The varying strength of U.S. contractions and expansions has also been studied by, e.g., Neftci (1984),
DeLong and Summers (1986), McQueen and Thorley (1993), McKay and Reis (2008), and Morley and
Piger (2012).
6
Table 1: U.S. Business Cycle Characteristics
Turning points
Peak date
Jan
May
Oct
Aug
May
Feb
Nov
Jul
Aug
Apr
Dec
Nov
Jan
Jul
Jul
Mar
Dec
1920
1923
1926
1929
1937
1945
1948
1953
1957
1960
1969
1973
1980
1981
1990
2001
2007
Duration
(months)
Trough date
Mar
Jul
Jul
Nov
Mar
Jun
Oct
Oct
May
Apr
Feb
Nov
Mar
Jul
Nov
Mar
Nov
Jun
1919
1921
1924
1927
1933
1938
1945
1949
1954
1958
1961
1970
1975
1980
1982
1991
2001
2009
P-value without outliers
Amplitude
(%)
Cumulation
(%)
Excess
(%)
T-to-P
P-to-T
T-to-P
P-to-T
T-to-P
P-to-T
T-to-P
P-to-T
10
22
27
21
50
80
37
45
39
24
106
36
58
12
92
120
73
15
11
10
40
10
5
8
7
5
7
8
13
3
13
5
5
15
20.07
45.81
23.21
20.87
72.25
99.52
18.38
38.06
19.54
19.61
55.29
21.05
25.95
5.91
29.00
40.58
14.37
-37.95
-18.92
-5.76
-72.11
-37.03
-33.65
-8.04
-8.85
-12.76
-6.25
-4.15
-13.57
-6.75
-8.91
-4.08
-3.47
-18.52
97.90
509.09
410.78
235.09
2023.92
4964.97
407.15
1049.35
519.24
325.38
3518.74
366.10
943.87
47.82
1731.95
2633.43
545.13
-366.46
-98.40
-21.65
-1742.67
-276.69
-105.90
-46.65
-62.08
-52.63
-33.23
-12.86
-61.14
-20.29
-77.69
-15.17
-14.91
-147.67
-1.25
-0.81
3.18
0.26
3.63
11.68
1.56
3.86
3.29
3.34
5.29
-0.65
3.07
0.78
4.17
1.48
0.18
-4.19
1.38
1.00
-6.61
-7.30
-0.99
-1.31
-3.81
-2.87
-1.18
0.73
2.60
-2.26
-1.18
-0.58
-0.90
0.03
0.50
0.22
0.30
0.29
0.18
0.19
0.07
0.24
Note: NBER peak (P) and trough (T) dates of the U.S. business cycle and corresponding business
cycle characteristics (duration, amplitude, cumulated movements and excess cumulated movements).
The outlier phase characteristics are shaded in grey. We report the p-values of Silverman’s modality
test for the subsamples without outliers.
of the four business cycle characteristics. Recall that multimodality of the Bootlier Plot
indicates the presence of one or more outlying observations. The Bootlier Plots of the
excess measure of expansions and of the duration, amplitude and cumulated movements of
recessions exhibit more than one mode. Outlier detection tests performed on the basis of
these plots lead to p-values smaller than 1%, indicating a rejection of the null hypothesis
of unimodality (no outliers). We iteratively run outlier detection tests to determine the
subsamples that are free of outliers. Business cycle phases with outlying characteristics
are shaded in grey in Table 1. We detect six outliers via our iterative procedure, one in the
excess cumulated movements of expansions, one in recession durations, one in recession
amplitudes, and three in the cumulated movements of recessions. At the bottom of Table
1 we also report the p-values of the outlier detection test for the subsamples with unimodal
Bootlier Plots. As Figure 3 shows, each Bootlier Plot becomes unimodal once the outliers
are removed from the samples.
The outlying phase characteristics correspond to four extraordinary episodes in the
history of the U.S. business cycle. The expansion that corresponds with an outlying excess
measure is relabeled as a boom, while the three recessions that exhibit outlying characteristics are classified as depressions. The first depression occurred between January 1920
- July 1921. Friedman and Schwartz (1963) argue that this deflationary downturn may
have been triggered by a negative aggregate demand shock partly caused by restrictive
7
Figure 3: Bootlier Plots
Recession
Full Sample
Without Outliers
0.4
kernel density
0.3
0.2
0.1
3.5
1.2
3.0
1.0
0.8
0.6
0.4
0.2
-3
-2
-1
0
1
2
3
4
5
6
kernel density
kernel density
0.01
0.02
0.03
0.04
kernel density
kernel density
0.5
0.4
0.3
0.2
0.1
0.0
-0.8 -0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
kernel density
kernel density
kernel density
1.2
1.6
2.0
0.0
-2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 2.0
2.4
bootstrap MTM statistics
120
100
40
30
20
10
0.01
80
60
40
20
0
-0.024
0.02
3.2
2.8
2.4
2.0
1.6
1.2
0.8
0.4
0.0
-1.2
0.007
-0.016
-0.008
0.000
0.008
500
400
300
200
0
-0.003 -0.002 -0.001 0.000
0.001
0.002
0.003
bootstrap MTM statistics
24
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
20
16
12
8
4
0
-0.10 -0.08 -0.06 -0.04 -0.02 0.00
0.02
bootstrap MTM statistics
400
300
200
100
0
-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003
bootstrap MTM statistics
Note: Bootlier Plots graphically represent kernel estimates of the bootstrap density of MTM statistics
for each business cycle characteristic. If the full-sample Bootlier Plot is multimodal, we iteratively
remove observations from the tails of the sample until the subsample Bootlier Plot becomes unimodal.
The left (right) panel shows full-sample and subsample Bootlier Plots for expansions (recessions).
The four rows correspond with duration, amplitude, cumulated movements, and excess cumulated
movements, respectively.
8
0.016
bootstrap MTM statistics
500
600
100
bootstrap MTM statistics
0.8
bootstrap MTM statistics
700
0.005
0.4
50
bootstrap MTM statistics
0.003
1.0
bootstrap MTM statistics
0.6
0.001
0.0
0
-0.05 -0.04 -0.03 -0.02 -0.01 0.00
0.05
0.7
-0.001
1.5
60
bootstrap MTM statistics
320
280
240
200
160
120
80
40
0
-0.003
2.0
bootstrap MTM statistics
bootstrap MTM statistics
40
35
30
25
20
15
10
5
0
-0.02 -0.01 0.00
2.5
0.5
0.0
-0.8 -0.4
kernel density
0.0
1.4
kernel density
kernel density
0.5
kernel density
Expansion
Full Sample
Without Outliers
0.04
Figure 4: Depressions and Booms of the U.S. Business Cycle
(a) The Great Depression
(b) The World War II Boom
Note: Figure (a): NBER peak-to-trough depression between August 1929 and March 1933. Figure (b):
NBER trough-to-peak boom between June 1938 and February 1945. The excess cumulative movements
are shaded in grey.
monetary policy between 1920 and 1921. The second depression is the August 1929 March 1933 collapse in real economic activity known as the Great Depression. Figure 4
(a) plots log industrial production from peak to trough. Our results reveal that the Great
Depression corresponds to the longest, sharpest, and most abrupt decline in real output,
with an amplitude of -72.1% and a cumulative loss of 1742.7% over 40 months. These values are between three to four standard deviations larger than for ordinary recessions. The
vast literature on the Great Depression typically describes an adverse interplay between a
contraction in aggregate production, systemic banking crises, and tight monetary policy
(see e.g., Bernanke, 1983). The third depression that we detect is dated between May
1937 and June 1938. Velde (2009) argues that the 1937-38 episode is a prime example
of a double-dip recession triggered by premature policy tightening in the aftermath of a
severe recession. Despite multiple comparisons with the Great Depression, the 2007-09
Great Recession is not classified as a depression by our statistical procedure. However,
it is undoubtedly the longest (15 months) postwar recession, with the largest amplitude
(-18.5%) and cumulative output loss (147.7%).6
The June 1938 - February 1945 wartime expansion constitutes the only genuine boom
phase. The WWII boom exhibits by far the largest excess cumulative movements: 11.7%
compared to a mean of 2.5% and standard deviation of 3.06%. Figure 4 (b) shows log
industrial production from trough to peak. The grey area in the plot corresponds to the
measure of excess cumulative movements. This measure reflects the departure of the real
output series from a triangular path for which the transition between two consecutive
turning points would be linear, it thus conveys information about the shape of the busi6
Notice that the outlier detection procedure does not directly compare the Great Recession with
the identified depression episodes. Whether the Great Recession is a depression solely depends on its
characteristics compared to other recessions, since the sequential omission of the depression characteristics
in the testing algorithm is equivalent to ultimately considering a subsample that does no longer contain
the three depression periods. Thus, if the Great Recession was a depression, it should lead to a multimodal
Bootlier Plot for this latter subsample, and this is not the case.
9
ness cycle phase. The concave shape of the upswing reflects a sharp surge in aggregate
production related to the arms industry, the expansion of productive capacity through
government-owned, privately operated capital, and an increase in labor force participation in durable goods manufacturing during World War II (see Braun and McGrattan,
1993). As the impetus of the economic boom abates by the end of the war, the economy
approaches its peak at a rather subdued pace, giving rise to the strong concavity observed
in the figure.
3
Predicting the four-regime business cycle
3.1
Multinomial logistic regression
We examine by means of multinomial logistic regressions whether macroeconomic and financial variables reflect leading information on the four-regime U.S. business cycle. Based
on the outlier detection results discussed in the preceding sections, we map the binomial cycle variable into a multinomial variable Stmul that corresponds to the expansion
(Stmul = 0), recession (Stmul = 1), boom (Stmul = 2), and depression (Stmul = 3) regimes
of the business cycle. Let It represent the information set which contains the past history of an L-dimensional vector of exogenous variables, xt . In a four-state multinomial
logit model, the probability that the business cycle is in regime j = 1, 2, 3 at time t + m
conditional on It obeys a logistic distribution function:
′
Pr
mul
St+m
= j|It
exp(βj xt )
=
,
P3
′
1 + h=1 exp(βh xt )
(3)
where m = 3, 6, ..., 24 stands for the prediction horizon (expressed in months). The model
is identified by imposing the condition that the expansion regime (j = 0) is the reference
state and all coefficients are expressed relative to this regime. Hence, using the fact that
the state probabilities must sum to unity, the conditional probability of the expansion
regime is given by:
1
mul
P r St+m
= 0|It =
.
(4)
P3
′
1 + h=1 exp(βh xt )
The multinomial logit model nests the binomial model which has been used in much of
the preceding literature (e.g., Estrella and Hardouvelis, 1991; Estrella and Mishkin, 1998;
Kauppi and Saikkonen, 2008).
Estimation is done by maximum likelihood optimization.7 The log-likelihood for a
7
In the business cycle literature probit models are more commonly employed than logit models to predict recessions (e.g., Estrella and Hardouvelis, 1991; Estrella and Mishkin, 1998; Kauppi and Saikkonen,
2008; Christiansen, 2013). However, this approach is less preferable in the multinomial context because
evaluating the state probabilities involves calculating high-dimensional integrals of the multivariate normal distribution, which makes the computation of the maximum likelihood estimates very complex (see
McCulloch and Rossi, 1994). Therefore, we opt for the logistic approach. Nevertheless, binary probit
and logit outcomes for our data were found to lie very close to each other, which suggests that it does
not matter whether one assumes a normal density or a logistic density for sake of maximum likelihood
optimization.
10
time series consisting of T observations is given by:
log (L(β)) =
T X
3
X
mul
mul
1(St+m
= j) log(P r(St+m
= j|It )),
(5)
t=1 j=0
where 1(.) is the indicator function. We measure the model’s goodness-of-fit using McFadden (1974)’s pseudo-R2 . For time horizons of m > 1, the prediction horizon exceeds
the data frequency, which creates serially correlated logit disturbance terms by construction. We remedy for this overlapping data problem using heteroskedasticity and serial
correlation robust standard errors (see also Estrella and Mishkin, 1998).
We are interested in the effect of a ceteris paribus change in one of the right-hand-side
(RHS) variables on the response probability defined in Equation (3), i.e., the so-called
marginal effects.8 The lth marginal effect gives the change in the probability that the
business cycle is in state j at time t + m in response to a one unit increase of the lth RHS
variable relative to its mean value (l = 1, ..., L):
!
P3
′
mul
= j|It )
∂P r(St+m
β
exp(β
x
)
h,l
t
h
mul
,
(6)
= P r(St+m
= j|It ) βj,l − h=1
P
′
∂xl,t
1 + 3h=1 exp(βh xt )
where βj,l is the lth element of βh .
Within this multinomial framework, we would like to assess to what extent macroeconomic and financial variables that traditionally exhibit leading indicator properties for
recessions and expansions have a different information content for depressions and booms.
Anderson (1984) defines a pair of regimes as “indistinguishable” if the RHS variables xt
deliver the same prediction for both regimes. If so, the two regimes can be merged into
a single regime. More precisely, let β1,l denote the coefficients that correspond to the explanatory variables xl,t in the recession regime relative to the reference (expansion) state,
and let β3,l stand for the coefficients of the depression regime relative to the reference
state. The null hypothesis that recessions and depressions are indistinguishable with respect to xt is given by β1,l = β3,l for l = 1, ..., L, which can be tested using conventional
likelihood-ratio (LR) and Wald statistics. Notice that a rejection of Anderson’s indistinguishability hypothesis would provide further justification for a four-regime business cycle
classification emerging from our outlier detection procedure.
3.2
Logistic regression results
Potential leading indicator variables are selected in line with previous empirical literature
on business cycle prediction (see, e.g., the seminal paper by Estrella and Mishkin, 1998).
Taking into acount that our RHS variables need to be available from 1919 onwards, we end
up with the term spread (SP READ), the S&P 500 stock index returns (∆ log SP 500),
the inflation rate (∆ log P P I), and the growth rate of log industrial production (∆yt ),
8
The majority of related business cycle studies only report coefficient levels β in Equation (3) because
the sign of the coefficients and the marginal effects are identical in the binary framework. However, in
the multinomial model an explanatory variable’s marginal effect depends on all coefficients, which implies
that it does not necessarily exhibit the same sign as the coefficient on the corresponding independent
variable.
11
Figure 5: Time Series and the U.S. Business Cycle
5
40
4
30
S&P 500 returns (%)
term spread (%)
3
2
1
0
-1
20
10
0
-10
-2
-20
-3
-30
-4
1920
1930
1940
1950
1960
1970
1980
1990
-40
1920
2000
1930
(a) Term Spread
12
real output growth (%)
16
7.5
5.0
PPI inflation (%)
1950
1960
1970
1980
1990
2000
1990
2000
(b) S&P 500 Returns
10.0
2.5
0.0
-2.5
-5.0
8
4
0
-4
-8
-7.5
-10.0
1920
1940
1930
1940
1950
1960
1970
1980
1990
-12
1920
2000
(c) Inflation
1930
1940
1950
1960
1970
1980
(d) Real Output Growth
Note: Time series are expressed as monthly percentages between March 1919 and June 2009. NBER
recessions are shaded in grey.
all at the monthly frequency.9 Data are taken from the St. Louis Fed and from Goyal
and Welch (2008). In line with the business cycle data, the sample covers the period
from March 1919 until June 2009. Figure 5 shows the data together with the NBER
recessions. The figure already provides some casual evidence that the business cycle and
the macroeconomic and financial variables are related. The term spread seems to display
counter-cyclical dynamics. Although stock returns, inflation, and output growth do not
exhibit a clear-cut pattern in the wake of recessions, they tend to decline during the three
depressions identified.
We distinguish between binomial and multinomial models with either a single RHS
variable (univariate model) or multiple RHS variables (multivariate model). Tables 2-8
report estimation results for these four possible logit model specifications. The tables
have a comparable structure and information content. Each table considers prediction
9
The term spread is defined as the difference between 10-year U.S. Treasury bonds and 3-month
Treasury bills. As inflation variable, we consider the growth rate of the producer price index (PPI),
which is a potential leading indicator for real economic activity because it measures the change in selling
prices received by domestic producers. We also ran a robustness check with CPI inflation and our results
were nearly identical.
12
time horizons of m = 3 up to m = 24 months and reports marginal effects together with
robust standard errors and accompanying Z-statistics. Since all regressors are expressed
in percentages, the marginal effects can be interpreted as changes in the m-month ahead
probability of a recession, depression, expansion or boom, in response to a one-percent
rise of a RHS variable. The tables also report log-likelihoods and pseudo-R2 values.10
Table 2 reports estimation results for univariate binomial logistic regressions as a
benchmark for comparison with the multinomial results, as well as a robustness check of
earlier binomial business cycle studies. In line with previous evidence, we find that the
slope of the yield curve is an accurate predictor of future recessions for all considered
time horizons. Rising term spreads reduce future recession likelihoods in a statistically
significant way. Different theoretical explanations have been launched for this empirical
observation. The most popular ones are related to the stance of monetary policy and
the information content on expectations regarding future economic prospects reflected
in the term spread. Current monetary policy can have a simultaneous impact on both
the yield curve and future real activity. For example, an expansionary monetary policy
can jointly induce a decline in the short rate – leading to a steeper yield curve – and
stimulate future economic activity. Furthermore, the Rational Expectations Hypothesis
(REH) of the term structure of interest rates can also contribute to understanding the
empirical relation between the yield curve and the business cycle. According to the
REH, an upward (downward) sloping yield curve indicates that future short-term interest
rates are expected to rise (fall). Hence, given that short-term interest rates are typically
pro-cyclical, a positive (negative) term spread signals a future business cycle expansion
(recession).
We also consider S&P 500 stock index returns in the univariate logit model. The
Efficient Markets Hypothesis implies that current stock prices equal the present value of
the expected future dividend stream, which in turn reflects expectations about future real
economic activity. Thus, in line with what one would expect, rising stock index returns
significantly reduce recession likelihoods for time horizons up to one year ahead.
Turning to the significance of the macroeconomic variables in the univariate logit
regressions, rising real output growth significantly reduces the probability of future recessions for time horizons up to three quarters ahead. The significant outcomes reflect the
temporal persistence of economic activity. However, the sign of the coefficient reverses for
a time horizon of two years ahead, which suggests a boom-bust cycle in the data. Finally,
inflation seems to predict the business cycle only up to six months ahead. Interestingly, a
marginal rise in inflation leads to a significant drop in the recession probability up to two
quarters in the future. This result is in line with, e.g., Estrella and Hardouvelis (1991) but
it contradicts certain structural macroeconomic studies, e.g., Smets and Wouters (2007)
who find a negative correlation between current inflation and future output growth.
Summarizing the univariate binomial model outcomes, both financial and macro variables impact the recession likelihoods, but the term spread is by far the best (in-sample)
predictor of future real economic activity. First of all, the term spread is the only variable
that exhibits significant predictive power over all time horizons. Moreover, the absolute
10
The considered models are estimated over the entire sample period. We do not perform out-ofsample prediction exercises for the depression and boom phases of the business cycle because the number
of in-sample depressions and booms (four in total) is too low to make a credible out-of-sample prediction
assessment.
13
Table 2: Univariate Binomial Logistic Regressions
∂P r(St+m =1)
∂SP READt
Z-statistic
Pseudo R2
Log-likelihood
∂P r(St+m =1)
∂∆ log SP 500t
Z-statistic
Pseudo R2
Log-likelihood
∂P r(St+m =1)
∂∆ log P P It
Z-statistic
Pseudo R2
Log-likelihood
∂P r(St+m =1)
∂∆yt
Z-statistic
Pseudo R2
Log-likelihood
P r(St+m = 1|It ) = F (β0 + β1 SP READt )
m = months ahead
3
6
9
12
15
18
21
24
-0.059
(0.010)
−5.98a
0.034
541.493
-0.081
(0.010)
−8.21a
0.067
509.912
-0.077
(0.010)
−8.09a
0.062
508.471
-0.069
(0.009)
−7.37a
0.050
509.971
P r(St+m = 1|It ) = F (β0 + β1 ∆ log SP 500t)
m = months ahead
3
6
9
12
15
18
21
24
-0.014
(0.003)
−4.97a
0.035
541.144
-0.000
(0.002)
-0.080
0.000
546.498
0.002
(0.002)
0.970
0.001
541.378
0.005
(0.002)
2.37b
0.005
534.386
P r(St+m = 1|It ) = F (β0 + β1 ∆ log P P It )
m = months ahead
3
6
9
12
15
18
21
24
-0.0623
(0.014)
-4.37a
0.030
544.064
0.016
(0.011)
1.43
0.002
550.097
0.010
(0.011)
0.94
0.001
546.068
0.005
(0.010)
0.53
0.000
541.691
0.0001
(0.010)
0.09
0.000
537.053
P r(St+m = 1|It ) = F (β0 + β1 ∆yt )
m = months ahead
3
6
9
12
15
18
21
24
-0.075
(0.011)
−6.66a
0.106
500.941
0.004
(0.006)
0.660
0.000
550.963
0.008
(0.006)
1.250
0.001
545.739
0.011
(0.006)
1.86c
0.003
540.248
0.014
(0.006)
2.48b
0.005
534.297
-0.079
(0.010)
−7.58a
0.061
525.523
-0.013
(0.002)
−4.72a
0.031
542.547
-0.033
(0.013)
-2.50b
0.009
555.090
-0.052
(0.009)
−6.03a
0.056
528.805
-0.093
(0.010)
−8.86a
0.085
511.739
-0.012
(0.002)
−4.86a
0.023
546.257
-0.002
(0.012)
-0.14
0.000
559.128
-0.023
(0.007)
−3.30a
0.012
552.622
-0.094
(0.010)
−8.97a
0.088
506.967
-0.008
(0.002)
−3.43a
0.010
550.145
0.017
(0.012)
1.46
0.002
554.639
-0.003
(0.006)
-0.460
0.000
555.723
-0.089
(0.010)
−8.59a
0.079
507.791
-0.003
(0.002)
-1.190
0.001
550.493
Note: Estimation results of univariate binomial logistic regressions with SP READt , ∆ log SP 500t ,
∆ log P P It , or ∆yt on the right-hand side. The marginal effect (∂P r(.)/∂xl,t ) is the partial derivative
of the predicted probability with respect to the RHS variable evaluated at its mean. Robust standard
errors are in brackets and the superscripts a , b , and c denote significance at the 1%, 5% and 10% level,
respectively. Identification is achieved by normalizing with the coefficients of the expansion state. We
provide the model log-likelihood, and goodness-of-fit is measured by McFadden’s pseudo R-squared.
14
Table 3: Multivariate Binomial Logistic Regression
P r(St+m = 1|It ) = F (β0 + β1 SP READt + β2 SP 500t + β3 IN F Lt + β4 ∆yt )
m = months ahead
3
6
9
12
15
18
21
24
Z-statistic
-0.058
(0.009)
−6.21a
-0.012
(0.003)
−4.10a
-0.032
(0.014)
−2.36b
-0.065
(0.011)
−5.83a
-0.077
(0.010)
−7.50a
-0.011
(0.003)
−4.29a
-0.008
(0.013)
-0.66
-0.044
(0.008)
−5.37a
-0.091
(0.010)
−8.77a
-0.011
(0.002)
−4.94a
0.010
(0.011)
0.98
-0.019
(0.007)
−2.72a
-0.094
(0.011)
−8.91a
-0.008
(0.002)
−3.40a
0.017
(0.011)
1.64
0.001
(0.007)
0.16
-0.090
(0.010)
−8.63a
-0.003
(0.003)
-0.95
0.014
(0.011)
1.31
0.008
(0.007)
1.07
-0.083
(0.010)
−8.37a
0.000
(0.003)
0.05
0.007
(0.010)
0.64
0.012
(0.007)
1.68c
-0.080
(0.010)
−8.41a
0.003
(0.002)
1.08
0.000
(0.010)
0.04
0.016
(0.006)
2.33b
-0.074
(0.009)
−7.92a
0.006
(0.002)
2.52b
-0.005
(0.010)
-0.48
0.018
(0.006)
2.80a
Pseudo R2
Log-likelihood
0.175
-462.707
0.142
-480.348
0.117
-493.654
0.101
-499.453
0.083
-505.234
0.071
-507.683
0.069
-504.684
0.065
-502.060
∂P r(St+m =1)
∂SP READt
Z-statistic
∂P r(St+m =1)
∂∆ log SP 500t
Z-statistic
∂P r(St+m =1)
∂∆ log P P It
Z-statistic
∂P r(St+m =1)
∂∆yt
Note: Estimation results of the multivariate binomial logistic regression with SP READt , ∆ log SP 500t, ∆ log P P It , and ∆yt on the right-hand side.
The marginal effect (∂P r(.)/∂xl,t ) is the partial derivative of the predicted probability with respect to the RHS variable evaluated at its mean. Robust
standard errors are in brackets and the superscripts a , b , and c denote significance at the 1%, 5% and 10% level, respectively. Identification is achieved
by normalizing with the coefficients of the expansion state. We provide the model log-likelihood, and goodness-of-fit is measured by McFadden’s pseudo
R-squared.
15
Table 4: Multinomial Logistic Regression with Term Spread
P r(St+m = j|It ) = F (β0 + β1 SP READt )
m = months ahead
3
6
9
12
15
18
21
24
Z-statistic
0.031
(0.010)
3.13a
-0.055
(0.008)
−7.34a
0.026
(0.003)
7.35a
-0.002
(0.007)
-0.25
0.047
(0.011)
4.49a
-0.072
(0.008)
−9.39a
0.027
(0.004)
7.42a
-0.003
(0.008)
-0.32
0.055
(0.011)
4.97a
-0.082
(0.007)
−11.05a
0.029
(0.004)
7.49a
-0.002
(0.009)
-0.22
0.056
(0.011)
4.97a
-0.082
(0.007)
−10.88a
0.030
(0.004)
7.54a
-0.004
(0.010)
-0.41
0.052
(0.011)
4.65a
-0.079
(0.007)
−10.58a
0.030
(0.004)
7.58a
-0.004
(0.009)
-0.40
0.047
(0.011)
4.31a
-0.072
(0.007)
−10.46a
0.030
(0.004)
7.61a
-0.005
(0.010)
-0.52
0.046
(0.010)
4.50a
-0.062
(0.007)
−9.04a
0.031
(0.004)
7.63a
-0.015
(0.008)
−1.92c
0.039
(0.010)
3.99a
-0.049
(0.007)
−6.90a
0.031
(0.005)
7.66a
-0.021
(0.006)
−3.59a
Pseudo R2
Log-likelihood
LR: Bo ≡ Ex
Wald: Bo ≡ Ex
LR: De ≡ Re
Wald: De ≡ Re
0.033
-938.930
12.520a
12.190a
12.603a
12.256a
0.054
-917.079
12.121a
11.840a
24.255a
22.995a
0.075
-895.691
12.435a
12.167a
38.657a
35.618a
0.076
-890.054
13.462a
13.150a
34.557a
32.033a
0.068
-889.984
14.892a
14.502a
27.860a
26.188a
0.057
-892.180
16.450a
15.950a
18.881a
18.059a
0.047
-894.103
17.596a
16.975a
3.428c
3.378c
0.040
-892.523
19.322a
18.545a
0.216
0.216
∂P r(St+m =0)
∂SP READt
Z-statistic
∂P r(St+m =1)
∂SP READt
Z-statistic
∂P r(St+m =2)
∂SP READt
Z-statistic
∂P r(St+m =3)
∂SP READt
Note: Estimation results of the univariate multinomial logistic regression with SP READt on the right-hand side. The marginal effect (∂P r(.)/∂xl,t ) is
the partial derivative of the predicted probability with respect to the RHS variable evaluated at its mean. Robust standard errors are in brackets and
the superscripts a , b , and c denote significance at the 1%, 5% and 10% level, respectively. Identification is achieved by normalizing with the coefficients
of the expansion state. We provide the model log-likelihood, and goodness-of-fit is measured by McFadden’s pseudo R-squared. We also report the LR
and Wald test for the null hypothesis that all coefficients (except intercepts) associated with a given pair of business cycle regimes (boom (Bo) and
expansion (Ex) or depression (De) and recession (Re)) are equal (indistinguishability). The test statistics are χ2 (1) distributed under the null.
16
Table 5: Multinomial Logistic Regression with Stock Returns
P r(St+m = j|It ) = F (β0 + β1 ∆ log SP 500t )
m = months ahead
3
6
9
12
15
18
21
24
Z-statistic
0.014
(0.003)
4.50a
-0.007
(0.002)
−3.61a
-0.000
(0.002)
-0.05
-0.006
(0.001)
−4.75a
0.014
(0.003)
4.58a
-0.007
(0.002)
−4.01a
-0.001
(0.002)
-0.40
-0.006
(0.001)
−3.95a
0.013
(0.003)
4.75a
-0.006
(0.002)
−3.26a
-0.002
(0.002)
-0.95
-0.006
(0.001)
−4.72a
0.010
(0.003)
3.71a
-0.004
(0.002)
−2.56a
-0.002
(0.002)
-1.15
-0.004
(0.001)
−2.71a
0.005
(0.002)
2.11b
0.000
(0.001)
0.20
-0.003
(0.002)
-1.56
-0.003
(0.002)
−1.83c
0.002
(0.003)
0.99
0.002
(0.002)
1.02
-0.002
(0.002)
-1.35
-0.002
(0.002)
-0.94
-0.000
(0.003)
-0.02
0.002
(0.001)
1.37
-0.002
(0.002)
-1.15
0.000
(0.002)
0.06
-0.003
(0.003)
-1.24
0.004
(0.002)
2.80a
-0.002
(0.002)
-1.20
0.001
(0.002)
0.61
Pseudo R2
Log-likelihood
LR: Bo ≡ Ex
Wald: Bo ≡ Ex
LR: De ≡ Re
Wald: De ≡ Re
0.024
-947.329
0.725
0.725
6.787a
6.689a
0.021
-949.444
1.753
1.770
3.745b
3.772b
0.018
-950.831
3.737b
3.846b
5.856b
5.839b
0.009
-954.210
3.852b
4.025b
1.935
1.978
0.005
-950.140
4.244b
4.493b
3.384c
3.49c
0.003
-943.983
2.764c
2.903c
1.940
1.990
0.002
-936.551
1.722
1.793
0.180
0.181
0.004
-925.964
1.341
1.395
0.143
0.142
∂P r(St+m =0)
∂∆ log SP 500t
Z-statistic
∂P r(St+m =1)
∂∆ log SP 500t
Z-statistic
∂P r(St+m =2)
∂∆ log SP 500t
Z-statistic
∂P r(St+m =3)
∂∆ log SP 500t
Note: Estimation results of the univariate multinomial logistic regression with ∆ log SP 500t on the right-hand side. The marginal effect (∂P r(.)/∂xl,t )
is the partial derivative of the predicted probability with respect to the RHS variable evaluated at its mean. Robust standard errors are in brackets and
the superscripts a , b , and c denote significance at the 1%, 5% and 10% level, respectively. Identification is achieved by normalizing with the coefficients
of the expansion state. We provide the model log-likelihood, and goodness-of-fit is measured by McFadden’s pseudo R-squared. We also report the LR
and Wald test for the null hypothesis that all coefficients (except intercepts) associated with a given pair of business cycle regimes (boom (Bo) and
expansion (Ex) or depression (De) and recession (Re)) are equal (indistinguishability). The test statistics are χ2 (1) distributed under the null.
17
Table 6: Multinomial Logistic Regression with Inflation
P r(St+m = j|It ) = F (β0 + β1 ∆ log P P It )
m = months ahead
3
6
9
12
15
18
21
24
Z-statistic
0.039
(0.015)
2.55b
-0.006
(0.015)
-0.40
0.007
(0.007)
0.99
-0.040
(0.007)
-5.82a
0.018
(0.013)
1.36
0.007
(0.012)
0.61
0.004
(0.007)
0.55
-0.029
(0.007)
-4.42a
-0.003
(0.012)
-0.21
0.020
(0.009)
2.20b
0.000
(0.007)
0.06
-0.018
(0.007)
-2.63a
-0.018
(0.012)
-1.44
0.031
(0.010)
3.24a
-0.000
(0.007)
-0.01
-0.013
(0.007)
-1.80c
-.020
(0.012)
-1.59
0.032
(0.010)
3.31a
0.002
(0.007)
0.28
-0.014
(0.007)
-2.09b
-0.019
(0.012)
-1.52
0.029
(0.009)
3.15a
0.006
(0.008)
0.74
-0.016
(0.006)
-2.75a
-0.018
(0.013)
-1.43
0.029
(0.009)
3.31a
0.008
(0.008)
1.01
-0.018
(0.005)
-3.73a
-0.014
(0.013)
-1.13
0.024
(0.009)
2.65a
0.007
(0.007)
1.03
-0.018
(0.005)
-3.89a
Pseudo R2
Log-likelihood
LR: Bo ≡ Ex
Wald: Bo ≡ Ex
LR: De ≡ Re
Wald: De ≡ Re
0.038
-933.459
0.104
0.107
40.986a
30.370a
0.017
-952.812
0.054
0.054
23.785a
21.312a
0.007
-962.034
0.008
0.008
12.862a
13.446a
0.007
-955.796
0.047
0.047
11.534a
12.197a
0.008
-938.015
0.255
0.254
12.745a
13.599a
0.009
-947.039
0.832
0.839
15.016a
16.118a
0.011
-927.815
1.315
1.345
18.858a
20.050a
0.010
-920.492
1.212
1.242
16.693a
17.792a
∂P r(St+m =0)
∂∆ log P P It
Z-statistic
∂P r(St+m =1)
∂∆ log P P It
Z-statistic
∂P r(St+m =2)
∂∆ log P P It
Z-statistic
∂P r(St+m =3)
∂∆ log P P It
Note: Estimation results of the univariate multinomial logistic regression with ∆ log P P It on the right-hand side. The marginal effect (∂P r(.)/∂xl,t ) is
the partial derivative of the predicted probability with respect to the RHS variable evaluated at its mean. Robust standard errors are in brackets and
the superscripts a , b , and c denote significance at the 1%, 5% and 10% level, respectively. Identification is achieved by normalizing with the coefficients
of the expansion state. We provide the model log-likelihood, and goodness-of-fit is measured by McFadden’s pseudo R-squared. We also report the LR
and Wald test for the null hypothesis that all coefficients (except intercepts) associated with a given pair of business cycle regimes (boom (Bo) and
expansion (Ex) or depression (De) and recession (Re)) are equal (indistinguishability). The test statistics are χ2 (1) distributed under the null.
18
Table 7: Multinomial Logistic Regression with Output Growth
P r(St+m = j|It ) = F (β0 + β1 ∆yt )
m = months ahead
3
6
9
12
15
18
21
24
Z-statistic
0.055
(0.010)
5.21a
-0.046
(0.008)
−5.53a
0.018
(0.004)
4.74a
-0.027
(0.004)
−6.89a
0.031
(0.008)
3.89a
-0.025
(0.006)
−4.22a
0.017
(0.004)
4.35a
-0.023
(0.004)
−6.54a
0.010
(0.007)
1.590
-0.007
(0.004)
-1.530
0.011
(0.005)
2.37b
-0.014
(0.004)
−3.71a
-0.008
(0.007)
-1.180
0.006
(0.004)
1.142
0.010
(0.005)
2.20b
-0.008
(0.004)
−1.84c
-0.015
(0.008)
−1.88c
0.008
(0.005)
1.750c
0.010
(0.005)
2.12b
-0.004
(0.005)
-0.690
-0.018
(0.008)
−2.22b
0.010
(0.005)
1.99b
0.010
(0.005)
2.03b
-0.001
(0.005)
-0.250
-0.022
(0.008)
−2.66a
0.012
(0.005)
2.46b
0.010
(0.005)
2.15b
0.000
(0.004)
0.030
-0.027
(0.009)
−3.03a
0.015
(0.005)
3.18a
0.011
(0.005)
2.26b
0.001
(0.004)
0.150
Pseudo R2
Log-likelihood
LR: Bo ≡ Ex
Wald: Bo ≡ Ex
LR: De ≡ Re
Wald: De ≡ Re
0.074
-898.841
11.425a
12.304a
12.691a
11.881a
0.048
-923.124
12.560a
13.326a
18.241a
16.590a
0.014
-955.277
5.938b
6.405b
7.543a
7.641a
0.007
-956.439
7.378a
7.899a
5.268b
5.435b
0.005
-949.483
7.697a
8.216a
2.224
2.239
0.005
-941.182
7.532a
8.027a
1.293
1.285
0.007
-931.428
8.924a
9.509a
1.022
1.008
0.009
-920.729
10.511a
11.191a
1.356
1.324
∂P r(St+m =0)
∂∆yt
Z-statistic
∂P r(St+m =1)
∂∆yt
Z-statistic
∂P r(St+m =2)
∂∆yt
Z-statistic
∂P r(St+m =3)
∂∆yt
Note: Estimation results of the univariate multinomial logistic regression with ∆yt on the right-hand side. The marginal effect (∂P r(.)/∂xl,t ) is the
partial derivative of the predicted probability with respect to the RHS variable evaluated at its mean. Robust standard errors are in brackets and the
superscripts a , b , and c denote significance at the 1%, 5% and 10% level, respectively. Identification is achieved by normalizing with the coefficients of
the expansion state. We provide the model log-likelihood, and goodness-of-fit is measured by McFadden’s pseudo R-squared. We also report the LR
and Wald test for the null hypothesis that all coefficients (except intercepts) associated with a given pair of business cycle regimes (boom (Bo) and
expansion (Ex) or depression (De) and recession (Re)) are equal (indistinguishability). The test statistics are χ2 (1) distributed under the null.
19
Table 8: Multivariate Multinomial Logistic Regression
∂P r(St+m =0)
∂SP READt
Z-statistic
∂P r(St+m =0)
∂∆ log SP 500t
Z-statistic
∂P r(St+m =0)
∂∆ log P P It
Z-statistic
∂P r(St+m =0)
∂∆yt
Z-statistic
∂P r(St+m =1)
∂SP READt
Z-statistic
∂P r(St+m =1)
∂∆ log SP 500t
Z-statistic
∂P r(St+m =1)
∂∆ log P P It
Z-statistic
∂P r(St+m =1)
∂∆yt
Z-statistic
∂P r(St+m =2)
∂SP READt
Z-statistic
P r(St+m = j|It ) = F (β0 + β1 SP READt + β2 ∆ log SP 500t + β3 ∆ log P P It + β4 ∆yt )
m = months ahead
3
6
9
12
15
18
21
24
0.033
(0.009)
3.47a
0.011
(0.003)
3.76a
0.028
(0.023)
2.08b
0.045
(0.010)
4.35a
-0.054
(0.007)
−7.27a
-0.007
(0.002)
−3.30a
0.000
(0.011)
0.01
-0.044
(0.008)
−5.41a
0.024
(0.004)
6.70a
0.047
(0.010)
4.59a
-0.001
(0.003)
-0.23
-0.004
(0.012)
-0.31
-0.025
(0.008)
−2.97a
-0.063
(0.007)
−9.05a
0.003
(0.002)
1.71c
0.022
(0.008)
2.71a
0.011
(0.005)
2.24b
0.029
(0.004)
6.79a
0.042
(0.010)
4.32a
-0.004
(0.003)
-1.45
-0.002
(0.012)
-0.14
-0.028
(0.009)
−3.22a
-0.051
(0.007)
−7.19a
0.005
(0.002)
2.93a
0.018
(0.009)
1.99b
0.014
(0.005)
2.78a
0.029
(0.004)
6.82a
0.050
(0.010)
4.67a
0.012
(0.003)
4.19a
0.012
(0.012)
0.99
0.024
(0.008)
3.07a
-0.070
(0.008)
−9.08a
-0.007
(0.002)
−3.78a
0.008
(0.009)
0.96
-0.021
(0.006)
−3.91a
0.025
(0.004)
6.69a
0.057
(0.011)
5.16a
0.012
(0.003)
4.55a
-0.000
(0.012)
0.05
0.005
(0.007)
0.74
-0.081
(0.008)
−10.82a
-0.004
(0.002)
−2.70a
0.017
(0.007)
2.40b
-0.006
(0.005)
-1.16
0.027
(0.004)
6.64a
0.057
(0.011)
5.07a
0.009
(0.002)
3.67a
-0.006
(0.012)
-0.50
-0.013
(0.007)
−1.83c
-0.082
(0.008)
−10.72a
-0.003
(0.002)
-2.01b
0.021
(0.008)
2.83a
0.008
(0.004)
1.70c
0.028
(0.004)
6.63a
Note: See next page.
20
0.053
(0.011)
4.80a
0.004
(0.002)
1.79c
-0.006
(0.012)
-0.50
-0.018
(0.008)
−2.34b
-0.079
(0.008)
−10.36a
0.001
(0.002)
0.85
0.024
(0.008)
3.12a
0.008
(0.005)
1.71c
0.028
(0.004)
6.67a
0.048
(0.011)
4.45a
0.002
(0.002)
0.71
-0.004
(0.012)
-0.37
-0.021
(0.008)
−2.62a
-0.073
(0.007)
−10.30a
0.003
(0.002)
1.55
0.022
(0.008)
2.94a
0.010
(0.005)
1.87c
0.029
(0.004)
6.77a
Table 8: Multivariate Multinomial Logistic Regression - Continued
Z-statistic
-0.001
(0.002)
-0.44
-0.003
(0.007)
-0.38
0.016
(0.004)
4.40a
-0.003
(0.005)
-0.50
-0.004
(0.001)
−2.89a
-0.025
(0.006)
−4.05a
-0.018
(0.004)
−4.69a
-0.001
(0.002)
-0.75
-0.006
(0.007)
-0.81
0.015
(0.004)
4.24a
-0.003
(0.007)
-0.45
-0.004
(0.001)
−2.58a
-0.015
(0.006)
−2.46b
-0.018
(0.004)
−4.77a
-0.002
(0.002)
-1.29
-0.007
(0.007)
-1.04
0.010
(0.004)
2.42b
-0.002
(0.009)
-0.27
-0.005
(0.001)
−3.86a
-0.009
(0.007)
-1.25
-0.010
(0.004)
−2.37b
-0.002
(0.002)
-1.52
-0.008
(0.007)
-1.06
0.009
(0.004)
2.27b
-0.003
(0.009)
-0.36
-0.004
(0.002)
−2.46b
-0.008
(0.007)
-1.05
-0.004
(0.005)
-0.99
-0.003
(0.001)
−2.00b
-0.005
(0.007)
-0.71
0.008
(0.004)
2.16b
-0.002
(0.009)
-0.27
-0.003
(0.002)
−1.84c
-0.012
(0.007)
-1.77c
0.001
(0.005)
0.12
-0.002
(0.001)
−1.76c
-0.001
(0.007)
-0.16
0.008
(0.004)
1.88c
-0.003
(0.010)
-0.34
-0.002
(0.002)
-1.01
-0.017
(0.006)
−2.77a
0.003
(0.005)
0.70
-0.002
(0.001)
-1.55
0.001
(0.008)
0.09
0.008
(0.004)
1.90c
-0.013
(0.008)
−1.65c
0.000
(0.001)
0.04
-0.019
(0.006)
−3.39a
0.005
(0.004)
1.24
-0.002
(0.001)
-1.59
0.000
(0.007)
0.03
0.009
(0.004)
2.02b
-0.020
(0.006)
−3.40a
0.001
(0.002)
0.71
-0.017
(0.005)
−3.25a
0.005
(0.004)
1.36
Pseudo R2
Log-likelihood
LR: Bo ≡ Ex
Wald: Bo ≡ Ex
LR: De ≡ Re
Wald: De ≡ Re
0.140
-834.393
23.996a
23.963a
52.661a
39.953a
0.121
-852.227
26.089a
26.066a
49.638a
41.496a
0.107
-865.332
22.290a
22.318a
53.525a
46.066a
0.097
-869.637
24.612a
24.618a
49.370a
44.043a
0.086
-872.657
26.575a
26.530a
46.060a
42.218a
0.074
-876.007
25.996a
25.780a
37.487a
35.203a
0.066
-876.350
27.228a
26.845a
22.035a
21.731a
0.062
-871.475
29.928a
29.330a
16.251a
16.782a
∂P r(St+m =2)
∂∆ log SP 500t
z-statistic
∂P r(St+m =2)
∂∆ log P P It
Z-statistic
∂P r(St+m =2)
∂∆yt
Z-statistic
∂P r(St+m =3)
∂SP READt
Z-statistic
∂P r(St+m =3)
∂∆ log SP 500t
Z-statistic
∂P r(St+m =3)
∂∆ log P P It
Z-statistic
∂P r(St+m =3)
∂∆yt
Note: Estimation results of the multivariate multinomial logistic regression. The marginal effect (∂P r(.)/∂xl,t ) is the partial derivative of the predicted
probability with respect to the RHS variable evaluated at its mean. Robust standard errors are in brackets and the superscripts a , b , and c denote
significance at the 1%, 5% and 10% level, respectively. Identification is achieved by normalizing with the coefficients of the expansion state. We
provide the model log-likelihood, and goodness-of-fit is measured by McFadden’s pseudo R-squared. We also report the LR and Wald test for the null
hypothesis that all coefficients (except intercepts) associated with a given pair of business cycle regimes (boom (Bo) and expansion (Ex) or depression
(De) and recession (Re)) are equal (indistinguishability). The test statistics are χ2 (4) distributed under the null.
21
values of the marginal effects in the term-spread regressions and the pseudo-R2 exceed the
other variables’ marginal effects and pseudo-R2 values for the majority of considered time
horizons. Finally, the term spread is the only leading indicator of which (the absolute
value of) the marginal effects and the pseudo-R2 exhibit an inverted U-shape (reaching a
maximum value for the annual time horizon); all other marginal effects and pseudo-R2 ’s
monotonically decline with the time horizon. Hence, the slope of the yield curve seems
to have an “optimal” predictive horizon for recessions that lies between three and five
quarters.
We also run the multivariate version of the binomial logit regression as a robustness
check, see Table 3. The outcomes of the univariate regressions are generally confirmed.
The term spread continues to have predictive power for future recessions over the whole
forecast horizon and its marginal effects are significant up to two years into the future.
Similarly, stock market returns and real output growth preserve their predictive power
for horizons of three quarters and one year, respectively. However, the marginal inflation
effect is no longer significant for the 6-month horizon. The binomial logistic regression
results largely confirm the preceding literature on binary business cycle prediction (e.g.,
Estrella and Hardouvelis, 1991; Estrella and Mishkin, 1998; Hamilton and Kim, 2002;
Kauppi and Saikkonen, 2008).
Turning to the multinomial logistic regressions, univariate results are reported in Tables 4-7 for the term spread, S&P 500 returns, the inflation rate, and real output growth,
respectively, while multivariate results are summarized in Table 8. The marginal effects
are reported across the four business cycle phases and add up to 0 for identification
purposes. We also report testing outcomes (LR and Wald) for the null hypothesis that
the RHS variables exhibit coefficients that are equal across a regime pair (recession vs.
depression or expansion vs. boom). If the null hypothesis cannot be rejected, the two
regimes are indistinguishable and can be merged into one regime; in case of rejection, the
RHS variables provide different predictions across the two regimes (see Anderson, 1984).
The marginal effects of ordinary recessions and expansions are somewhat smaller (in
absolute value) in the multinomial regressions compared to the binomial results but remain
statistically significant for the same variables and time horizons. Hence, the binary results
are robust to adding regimes to the business cycle. Strikingly, however, the term spread
fails to predict depressions despite the fact that it preserves predictive power for recessions
and expansions. At the same time, pseudo-R2 values and marginal recession effects are
largest for the term spread across different time horizons. Just as in the binomial case,
the term spread is the only early warning indicator whose marginal effects and pseudo-R2
exhibit an inverted U-shape and a comparable optimal predictive horizon.
In contrast to the term spread, the depression probability significantly diminishes when
output growth, S&P 500 returns or inflation rise. Specifically, the univariate outcomes
show that output growth and stock returns exhibit predictive power for depressions up to
the 1-year time horizon. Remarkably, inflation seems to be the most important leading
indicator for economic depressions in terms of marginal effects and across time horizons
(up to 24 months). Moreover, the inflation rate has a significantly different information
content for recessions and depressions across all forecast horizons (see the outcomes of the
LR and Wald tests). A ceteris paribus one percent rise in inflation leads to a significant
increase in the recession probability between 9 and 24 months ahead, while it leads to a
significant drop in the depression probability at virtually every forecast horizon. The ap22
Figure 6: Predicted Probability of Recession and Depression (Multivariate)
1.0
.7
predicted probability of depression
predicted probability of recession
.8
.6
.5
.4
.3
.2
.1
.0
1920
1930
1940
1950
1960
1970
1980
1990
2000
0.8
0.6
0.4
0.2
0.0
1920
2010
(a) Probability of recession
1930
1940
1950
1960
1970
1980
1990
2000
2010
(b) Probability of depression
Note: Predicted probabilities of a recession (Figure (a)) and depression (Figure (b)) occurring three
months ahead. The probabilities are estimated from the multivariate multinomial logit model including
all leading indicators (SP READ, ∆ log SP 500, ∆ log P P I, and ∆y). Recessions are shaded in light
grey and depressions are shaded in dark grey.
parently different impact of a rise in inflation on the occurrence of recession vs. depression
regimes is somewhat puzzling but the existing literature gives some hints to what may
be behind this phenomenon. Using a New Keynesian model on postwar macroeconomic
data, Smets and Wouters (2007) show that inflation is primarily driven by price and wage
mark-up shocks, inducing a negative correlation between current inflation and future output. This is in line with our positive marginal recession effect for inflation. Notice indeed
that most recessions in our sample are situated after World War II. On the other hand,
the significantly negative effect of a rise in inflation on the depression probability is in
line with the observation that all three (prewar) depressions were deflationary episodes,
see Figure 5 (c).
As a robustness exercise, we report estimates of a multinomial logit regression that
includes all four explanatory variables in Table 8. The variables’ marginal effects preserve
the same signs and degrees of statistical significance. Moreover, LR and Wald tests
that compare coefficients across regimes lead to the same conclusions as the test results
obtained from univariate logit regressions.
Figure 6 complements the regression results with graphs of the three-month-ahead predicted probabilities of a recession and depression for the multivariate multinomial logit
model. These probability plots provide a qualitative complement to goodness-of-fit measures like the pseudo-R2 . The figure reveals that the leading indicators clearly differentiate
between ordinary and extraordinary business cycle phases. The peaks in the predicted
recession probability coincide with the recessions shaded in light grey, while the peaks of
depression probabilities are associated with depressions shaded in dark grey. The central message from the figure is that depression probabilities exceed recession probabilities
during depressions: they are higher than 80% in all three cases. Thus, the multinomial
23
model successfully disentangles economic recessions and depressions.11
4
Conclusions
The business cycle is traditionally approached as a sequence of recessions and expansions
in real economic activity. However, it is well-known that some recessions and expansions
have been much longer, deeper or more abrupt than others. Thus, the question arises
whether the binary characterization of the business cycle is not overly simplistic. Vast
contractions of real output have indeed stimulated interest in rare economic disasters
among economists. The binary framework may be overly restrictive for modeling such
rare and severe events.
Questioning the validity of the binary approach therefore constitutes the starting point
of this paper. We distinguish depression and boom episodes from ordinary recessions and
expansions by means of a novel nonparametric framework. Using industrial production
data and the peak and trough dates of the NBER business cycle between 1919 and 2009,
we compute the duration, amplitude, cumulated movements, and excess cumulated movements of each recession and expansion phase. A nonparametric outlier detection test is
applied to the empirical distribution of these business cycle characteristics. The outlier
test enables one to distinguish tail events in cycle characteristics arising from the same
probability distribution from outlying characteristics generated by a different underlying
process. We relabel recessions and expansions in U.S. business cycles as depressions and
booms provided at least one of the corresponding cycle characteristics exhibits outlying
behavior. The presence of outlying business cycle phases justifies the specification of an
econometric model that is richer in dynamics than the binary approach. The re-mapping
of the binary NBER business cycle into a four-regime cycle constitutes the first contribution of the paper. We identify four extraordinary episodes: three recessions are relabeled
as economic depressions and one expansion is classified as an economic boom. Interestingly, the 2007-09 Great Recession is not identified as a depression by our outlier test
procedure.
In the second part of the paper, we use multinomial logistic regressions to analyze
whether the four-regime business cycle can be predicted using macroeconomic and financial variables. Several key results emerge from our logistic regression analysis. First, we
find that recessions become less likely when the term spread, stock market returns, and
real output growth increase, regardless of whether one considers binomial or multinomial
business cycle phases. Stated otherwise, the predictive power that macroeconomic and
financial variables exhibit towards recessions in a binary prediction framework carries over
to the multinomial framework. Second, using statistical tests within the multinomial logit
framework, depressions are found to be significantly different from recessions. This provides further justification for the four-regime business cycle classification emerging from
our outlier detection procedure. Finally, although the slope of the yield curve outperforms
other leading indicators in predicting recessions, it does not have anything to say about
11
Interestingly, the indicators signal an impending depression with a probability of about 80% in March
2009, which suggests that the U.S. economy was evolving towards a depression regime at that time.
However, the depression probability drops to approximately 10% by June 2009; hence, the economy
avoided slipping into depression.
24
future economic depressions. In contrast, a decline in stock market returns, real output
growth, or inflation increases the likelihood of a depression.
A
Appendix: Business cycle characteristics
Formally, we observe i = 1, ..., n business cycle phases over a given period t = 1, ..., T
(n < T ). First, the duration of the ith recession (expansion) measures the number of
months between a peak (trough) occurring at t1 and the next trough (peak) occurring at
t2 :
t2
X
Di =
Qt ,
t=t1
Stbin
where Qt =
in case of a recession and Qt = 1 − Stbin in case of an expansion. Second,
the amplitude of the ith recession (expansion) measures the change in real output yt
between a peak (trough) at t1 to the next trough (peak) at t2 :
Ai = yt2 − yt1 .
Third, the cumulative movements of yt within the ith phase measure the overall cost
(gain) of the recession (expansion):
Ci =
t2
X
(yj − yt1 ).
j=t1 +1
Finally, the excess cumulative movements measure the curvature of yt within the phase
defined as:
Ei = (Ci − 0.5Ai − 0.5Ai Di )/Di .
The excess cumulated movements provide an approximation to the second derivative of
the time series. If Ei > 0, the business cycle phase exhibits a concave shape, i.e., the
slope of yt changes abruptly at the beginning of the phase but the changes in slope become
more gradual as the phase approaches its turning point. Conversely if Ei < 0, the shape
is convex, and the slope changes more gradually at early stages of the phase (see Camacho
et al., 2008).
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